Re: Dedekind Cuts, Fundamental Sequences: why?

Glen Wheeler wrote:

"Hatto von Aquitanien" <abbot@xxxxxxxxxxxxxx> wrote in message
news:2NWdne8gQLa8lvTbnZ2dnUVZ_u3inZ2d@xxxxxxxxxxxxxxxx
Dave Seaman wrote:

On Thu, 07 Jun 2007 07:54:50 -0400, Hatto von Aquitanien wrote:

I guess one could argue that even if we do have Cauchy sequences in
which
some of the representations have irrational elements, they must all
have representations which are purely rational due to the fact that the
real
numbers were completely defined by the latter. That is, of course,
assuming that such a construction actually took place.

Yes, a sequence of reals has a representation as a sequence of sequences
of rationals.

Is such a sequence of sequences of rational numbers a real number?


A sequence of sequences is exactly that. A sequence of sequences. I
suspect you want to talk about the limit. In that case, if it exists it
will be a real number. It might also be rational.


"7. The [set of] equivalence classes in R = C/I that consist of
positive sequences [is] closed with respect to addition and
multiplication, and [satisfies] the trichotomy law."[*]

I believe that I am to understand the set of equivalence classes in R as
defining the set of of real numbers. So in this formulation real numbers
are neither limits, nor are they sequences of rational numbers. They are
equivalence classes of sequences of rational numbers.

It seems this is the kind of thing that Weyl didn't like. It's hard to
really glean his intent and put it into clear words, but at the risk of
being very wrong, I will say that Weyl wanted the definition of real
numbers to be as intuitively obvious as that of natural numbers. I'm not
sure Weyl was wrong about the existing definitions being circular, but I do
have the sense that Weyl's reasoning may have been circular. That is, he
seems to have wanted the definition of the reals to agree with his
intuitive notion of the reals which existed prior to any formal definition.

I seriously doubt that I am the first person to raise this following
objection: If we assume by axiom that for x,y \in \R => x<y xor x=y xor
x>y, then x!=y => x-y!=0. We also have the possibility that x \in \Q and
y !\in \Q, which implies x!=y, and thus x-y!=0. Now, suppose I have some
series which converges in finite time to a real value v !\in \Q. For every
member of any Cauchy series in the equivalence class representing v, we can
find z \Q such that |v-z| > 0.

This is the kind of thing which makes me feel as though logic alone is
insufficient to prove that a sequence of rational numbers can converge to
an irrational real number.

[*]Please let me know if my modifications are inappropriate.
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